CARROLLTON, Ga.—Brandi Shirey wants to borrow at least $20,000 to expand the birthday- and wedding-cake business she started four years ago after leaving her job as a bookkeeper.

Demand for the cakes, which sell for $150 to $500, overwhelms her home kitchen. She plans to use about $2,000 from her savings to move into a nearby storefront next month. But the 28-year-old Ms. Shirey believes her credit record and financial paperwork have...